Rational Definition and 616 Threads

Hi all, During my research i ran into the following general type of equation: \exp(ax+b)=\frac{cx+d}{ex+f} does anyone have an idea how to go about solving this equation? thx in advance

Find the partial fraction decomposition for the rational function. $$\frac{-4x^2 - 8x - 19}{(x^2 + 2)(x-9)}$$ I'm not sure what to do.

1. Prove that \sqrt{2} is not a rational number using the rational zero theorem 2. Homework Equations : x2-2 The Attempt at a Solution So, I was thinking to just show that the only rational numbers that have the form \frac{c}{d} are \pm 1, \pm 2, but \sqrt{2} is not any of those...

Hi I have the function I have worked out the domain as not equal to 1 using the fact the denominator can't equal 0. Now I am stuck finding the range. How would I find it without graphing. The answer is y<0 , y=>3.

referring to the photo attached, can anyone expalin and give suitable numerical example please? i couldn't understand the notes.

form quadratic equation $ax^2 +bx+c=0$ in parametric form such that a,b,c are integers in AP and it has got rational roots

5c) Simplify. $$\frac{2x}{3y} - \frac{x^2}{4y^3} + \frac{3}{5y^4} $$This is what I did, which is wrong according to the textbook. Could someone point out what I did wrong and how to correct it? Thanks. $$\frac{(2x)(4y^3)-(x^2)(3y)}{(3y)(4y^3)} + \frac{3}{5y^4}$$ $$\frac{8xy^3-3x^2y}{(12y^3)}...

7a) Simplify and state any restrictions on the variables: $$\frac{x^2-5xy+4y^2}{x^2+3xy-28y^2} ⋅ \frac{x^2+2xy+y^2}{x^2-y^2} $$ I'm not really sure what a good process would be to simplify this. This is what I tried to do, (below) which is wrong. Could anyone point out what I did wrong and...

Please see attached. The question is to determine f(0). In my opinion the question wants me to find f(x).. If f(x)=1, then 1. f:[0,1] -> R is continuous (do I need to prove?) 2. Obviously, f(1) = 1 3. f(x) is a rational number for all xε[0,1] so f(0) = 1 Am I correct?

Is it true that in some cases an irrational number raised to a rational power is rational? If so what kinds of irrational numbers?

Q4.) Quadratic equation: $$\frac{-2x^2-14x-49}{x^3-7x^2}$$

I've been dealing with several integrals involving rational functions. I have encountered myself arriving to an integral that requires the application of the following recursive formula: \int\frac{1}{(u^2+α^ 2)^m} \, du= \frac{u}{2 α^ 2 (m-1)(u^2+α^ 2)^{m-1}}+\frac{2m-3}{2 α^2...

Homework Statement Use the rational roots theorem to prove 31/2-21/3is irrational. The Attempt at a Solution My teacher strongly hinted to us that this problem had something to do with the fact that complex roots come in conjugate pairs, and all we had to do was, "flip the sign"...

I have a question why is $$\frac{2x^2-1}{x^2+1}$$ equal to $$2-\frac{3}{x^2+1}$$ ??

On The graph, the following information was given: Point of discontinuity( -2,0) and (1,-3) X intercept at ( 2,0) Yintercept (0,-4) This is what I had : (x+2)(x-2)(x-1)/ (x+2)(x-1) But the answer is that u have to put (x+2)^2 in the numerator instead of what I had. But I thought...

Hi, could someone please help me with this one: I'd need to form a sequence of rational functions ##R_{n}(x)## such that ##\lim_{n \to \infty} R_{n}(x)=\theta(x)##, where ##\theta(x)## is the Heaviside step function. The functions ##R_{n}(x)## should preferably be limited in range, i.e. for some...

this is not a homework question, but rather I feel like there is a contradiction in the theorem and just want clarity. I know the theorem is correct so I am looking for help in where the mistake is in my logic. take f(x) = x^3 + x^2 - 4x- 7 the rational roots theorem says if there are any...

Homework Statement Hello, I know the direct substitution property in calculus. But, the definition of a rational function still confuses me. For example, are these rational functions: y=(x^2+2x+1)/(x+1) y=((x^2+2)^(1/2))/(x+1) The denominator of the first one could cancel. So...

Homework Statement Solving formulas involving Rational expressions for each excersise, solve for the indicated variable Homework Equations (x-y)/xy = z Solve for x The Attempt at a Solution I started by multiplying both sides by xy, to which I got (x-y) = xyz Then I tried multiplying...

Hi, I am in a first semester Calculus I course in college with an intermediate skill level with precalc and a basic understanding of limits and infinity. I do not understand how to solve this problem I attempted to do so only to find out after completion that ∞/∞ is indeterminate rendering my...

Homework Statement Identify the solution set of the inequality. Homework Equations 5x + 1 / x- 1 ≥ 7 The Attempt at a Solution I multiplied both sides by x - 1 which gave me 5x+1 ≥ 7x - 7 Then I combined like terms together which gave me 8 ≥ 2x I divided both sides by 2 which gave...

Homework Statement I have the equation $$ a^3+2b^3+4c^3-6abc=0 $$ and I want to show that for ##a,b,c\in\mathbb{Q}## and ##a,b,c\not=0## that there are no solutions. The Attempt at a Solution I tried many different approaches but this was the only one that seemed to work. I am not sure...

Homework Statement I want to find the transition matrix for the rational canonical form of the matrix A below. Homework Equations The Attempt at a Solution Let ##A## be the 3x3 matrix ##\begin{bmatrix} 3 & 4 & 0 \\-1 & -3 & -2 \\ 1 & 2 & 1 \end{bmatrix}## The...

Homework Statement Find continuity of function f(x)= (x^2-1)/(x-1) at x = 1 Homework Equations limit f(x) as x-> = L The Attempt at a Solution i KNOW it can be easily solved by stating that at x = 1 function becomes infinity,so discontinous it is actually...But as we do in finding domain...

The newest challenge is the following: As an example, we can easily go from ##0## to ##-1/3##. Indeed, we can apply ##T## to ##0## to go to ##1##, we apply ##T## to go to ##2##, we apply ##T## to go to ##3##, and then we apply ##R## to go to ##-1/3##.

Hello, I am trying to solve the next integral, using this method of rational functions. My problem is, that I turn the function into two expressions with the same denominator...when I try to use values of x, I will always make zero of both sides, if you know what I mean \[\int...

Here is the question: I have posted a link there to this topic so the OP can see my work.

I'm wondering if Pi is a rational number in any other base besides base Pi. Also is there a formula or function to figure this out? I'm not what the relevance would be if we could find one since the integers might be irrational if we did but I am just curious if indeed Pi is only a rational in...

Explain why the sum, the difference, and the product of the rational numbers are rational numbers. Is the product of the irrational numbers necessarily irrational? What about the sum? Combining Rational Numbers with Irrational Numbers In general, what can you say about the sum of a rational...

Hello everyone, I am a bit confused about definitions rules. I can have more questions but for now I want to ask only one question: Let us say I have a number: \sqrt[6]{3x3x3x3x3x3} 3x3x3x3x3x3 is equal to both 27^2 and (-27)^2. But If I write these two expressions separately I can get...

In general I am rather confused by this type of problem. The textbook has a single example and does not show (m)any of its steps so I'm lost. I have a test this coming Thursday and the following is the only question of this type that the prof. has recommended: "23. Use equations (12) and (13)...

Is there a way to make these a polynomial? 1)$$\sqrt{A}+\sqrt{B}+\sqrt{C}=D$$ 2)A1/3 + B1/3 = C Or how else can it be solved?

you can list and match up all rational numbers with irrational numbers this way.. lets say i have an irrational number 'c'. Rational->Irrational r1->cr1 r2->cr2 . . . rn->crn There exists an irrational number that is not on this matching, (not equal to any of the crx's) this...

Homework Statement I came across this integral recently while tutoring: ##\displaystyle \frac{1}{5} \int \frac{-x^3+2x^2-3x+4}{x^4-x^3+x^2-x+1}~dx## Homework Equations The Attempt at a Solution I'm not sure how to approach this. At first I suspected partial fraction decomposition...

I have found some trouble in trying to prove this question.please help mw with that. Q1) If (a+b)/2 is a rational number can we say that a and b are also rational numbers.? Justify your answer. I have tried the sum in the following way. Assume (a+b)/2=p/q (As it is rational) Lets...

1) integral (upper bound:1, lower bound:0) (x^2+1)/(x^3+x^2+4x) dx 2) integral (upper bound:1, lower bound:0) (x^4+x^2+1)/(x^3+x^2+x-3) dx Now I know how to use Partial Fractions,My question is: 1) For the first part ln(x) is not defined at 0 ¼ʃ1/x dx + ¼ʃ(3x-1)/(x²+x+4) dx = ¼ ln|x| +...

Homework Statement x4 - 4(x3) + 3(x2) -2x +1 = 0 Homework Equations Rational Root Theorem, q/p The Attempt at a Solution Hello everyone. Today, I've learned the rational root theorem( it's a bit late, isn't it? :( ) and thus wanted to see how it works. According to the...

There are lots of examples of numbers where "is it a rational number" has been an open question for a while before being proved as not being rational. Pi and e being famous examples. Some of them are still open, like pi+e, or the Euler-Mascharoni constant, but I think the general consensus is...

Homework Statement Integrate (8-16x)/(8x^2-4x+1) dx Q Homework Equations I separate it first 8/(8x^2-4x+1)dx -16x/(8x^2-4x+1)dx Then I have no idea what to do next. The Attempt at a Solution none ;(

I am struggling to draw this point home: To prove that R has LUB property, we used the following reasoning: First we defined R to be set of cuts (having certain properties) where each cut corresponds to a real number and then we proved any subset A of R has LUB (least upper bound) property...

Let $r(x)\in\mathbb Q(x)$ be a rational function over $\mathbb Q$. Assume $r(n)$ is an integer for infinitely many integers $n$. Then show that $r(x)$ is a polynomial in $\mathbb Q[x]$.

Here is the question: Here is a link to the question: How to find the intervals of increase and decrease? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Can someone explain how these are equivalent. sqrt((-3)^2) = (-3)^2/2 =sqrt(9) and (-3)^1 3 is not equal to -3 (-3)^2/2 can be expressed as: (-3^2)^1/2 and (-3^1/2)2 (9)^1/2 and (sqrt(-1)sqrt(3))^2...

We cannot put the ratio of circumference/diameter in the form p/q. In this case the circumference. Because any number of sides of a regular polygon perimeter to calculate the circumference will not fit to the circumference of a circle. That is the number of sides of a polygon tends to...

Hi there, I've been learning about Solving for x in rational expressions with Mono, Bi and tri nomial denominators, and I was curious if there was an easier way to solve them than by factoring. I already understand and fully grasp the concept, but its just annoying to go through all the same...

Hi there. I am currently taking "College Math 1" at the local CC and I have encountered something that confuses me regarding rational expressions and their domains. The definition given by the textbook for rational expressions is: "the set of real numbers for which an algebraic expression is...

Homework Statement I'm studying for my final exam and came across this problem: Let f and g be entire analytic functions and |f(z)|<|g(z)| when |z|>1. Show that f/g is a rational function. The Attempt at a Solution I really have no clue where to go :(

Homework Statement ((a^3-b^3)/(a^2-2ab+b^2))/((2a^2+2ab+2b^2)/(9a^2-9b^2)) Not using complex number system. Not concerned with domain. Find the quotient and put it in simplest terms. Homework Equations The Attempt at a Solution Too many to transcribe. Apparently I'm missing...

Here is the question: Here is a link to the question: Polynomials, please help 10 points!? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Simplify $$\frac{x^2-4x+3+(x+1)\sqrt{x^2-9}}{x^2+4x+3+(x-1)\sqrt{x^2-9}}$$ where $$x>3$$.